Transport properties in artificial lateral superlattice

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Superlattices and Microstructures, Vol. 16, No.3, 1994

TRANSPORT PROPERTIES IN ARTIFICIAL LATERAL SUPERLATTICE K.Tsukagoshi, S.Wakayama, K.Oto, S.Takaoka, and K.Murase Department of Physics, Faculty of Science, Osaka University 1-1 Machikaneyama, Toyonaka, 560, Japan K.Gamo Department of Electrical Engineering, Faculty of Engineering Science and Research Center for Extreme .Materials, Osaka University 1-1 Machikaneyama, Toyonaka, 560, Japan ( Received 22 August 1994 ) We investigate commensurate oscillations in ordered and disordered artificial lateral superlattice (ALS) systems, in which the anti-dots are arranged in a square or triangular lattice. With increasing disorder of the anti-dot location, the peaks of the commensurate oscillations fade out. The peak heights are more strongly affected by the disorder along perpendicular direction to the current than by that along the parallel direction. In the square ALS system, the commensurate oscillations seem to be determined principally by the order along the perpendicular direction to the current, while in the triangular ALS system, the commensurate oscillations would be determined by the nearest neighbor distance between anti-dots and the order along the perpendicular direction. In addition, the appearance of each peak is determined by the ratio of the anti-dot diameter to the ALS period. The weak localization effect in very low magnetic field and the strongly temperature dependent conductance in the absence of magnetic field are also observed in the ALS systems.

1

Introduction

In the two dimensional electron gas (2DEG) subjected to an artificial lateral superlattice (ALS) system, which consists of "anti-dots" with submicrometer periods, the peculiar magnetoresistance oscillations have been observed. The oscillation peaks appear in the magnetoresistance when the diameters of the cyclotron orbit are commensurable with the period of the anti-dot lattice. They are called "commensurate oscillations". They are interpreted by a model of pinned electron orbits around a group of anti-dots [1,2] or the classical chaotic dynamics [3]. The mechanism of the oscillations, however, are still open to discussion. Moreover, the negative magnetoresistances and strongly temperature dependent resistances have been observed in various ALS systems at low temperature [4-9J. Some of these experiments have been made in the systems produced by focused ion-beam (FIB) techniques. Since the energy of FIB was so high that the damage around the anti-dots could not be avoided, the anti-dot diameter was large enough to be comparable to the period. The boundary scattering of electrons by the

0749-6036/94/070295 + 07 S08.00/0

anti-dot walls became prominent. The negative magnetoresistances have been interpreted as an Anderson localization by the electron-electron scattering and the one-dimensional weak localization by the boundary scattering [7,10]. In other ALS systems [8,9], the negative magneto resistance was explained by a numerical calculation based on a classical model in which the external electric field and Hall field were explicitly taken into account. In this paper, we investigate the commensurate oscillations in ordered and disordered ALS systems, especially only one-direction disordered ALS systems in which the disorder is introduced either to the parallel or to the perpendicular direction to the current, and the expanded or the reduced ALS systems' It is found that the commensurate oscillations in square ALS systems is determined by the order along the perpendicular direction to the current, and the oscillations in triangular ALS systems is determined by the nearest neighbor distance of the anti-dots in addition to the order along the perpendicular direction. We also study the dependence of the ALS period and the anti-dot diameter. In addition, the negative magnetoresistance at very low magnetic field is investigated.

© 1994 Academic Press Limited

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Superlettices and Microstructures. Vol. 16. No.3. 1994

2

Fabrication and Measurement

Our devices were fabricated from a wafer of a GaAs/ AlGaAs heterostructure. After drawing triangularly or squarely arranged points [diarnetereed, period=a) on the polymethylmethacrylate (PMMA) resist which was coated on the mesa-etched Hall bar, the holes were etched by 1 keY Ar+ ion milling. Thus, the 2DEG under the etched holes were depleted, and the "antidots" were constructed. In the same device, there are four or five anti-dots array regions (Fig.l(a)) in order to make little difference of the fabrication process between each ALS system. The carrier density and the elastic mean free path of the 2DEG region in the device are 5.2 x 1011 ern"? and 12 p.m respectively at 1.5 K and under illumination. Magnetic field 8 was applied along the perpendicular direction to the plane of the 2DEG. The magnetoresistance was measured by an AC resistance bridge at 15 Hz. The peak due to commensurate oscillations did not depend on the current between 3 nA and 1 p.A.

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3 3.1

Results and Discussion

Disordered ALS Systems

The magnetoresistance of the ordered and the disordered triangular ALS system is shown in Fig. l(b). Each ALS system has anti-dot arrays in the area of 200 p.m x 150 p.m. The shifts of the anti-dot location due to the disorder were introduced to all directions from the ordered anti-dot arrays [4] in the Gaussian distribution (standard deviation = 0-). The value of 0 was chosen below the distance of a. In the magnetoresistance of the ordered ALS system (0=0), the peaks due to the commensurate orbits clearly appear at the magnetic fields corresponding to the encirclement of the electron orbit around 1 or 7 anti-dots. With increasing 0, the peak heights of the commensurate oscillations decrease. Especially, the peak due to the encirclement orbit around 1 anti-dot almost disappear at 0=0.25a although there are enough space for the electrons to encircle around the anti-dot. It turns out that the commensurate oscillations can not be observed without the order of anti-dot location in the ALS system. Moreover, we measured the magnetoresistance of ALS systems where the disorder is introduced either to X-direction (X-disorder) or to V-direction (Ydisorder) (Fig.2). The X-direction is the current direction (Fig.2(a)(c)). In the triangularly arranged ALS systems (Fig.2(a)), the peak heights in the V-disordered ALS systems decrease more remarkably than those in the X-disordered ALS systems (Fig.2(b)). The peak heights are affected by the disorder along the V-direction more strongly. Since the X-direction and the V-direction are not originally equivalent in the triangular ALS sys-

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Magnetic Field (kG) Fig. 1. (a) Schematic view of sample. There are four or five ALS systems in the same sample. (b) Magnetoresistances of triangularly arranged ALS systems with various standard deviation (0). With increasing 0, the peak of the commensurate oscillations fades out.

tern, we also investigated the square ALS system where two distances were equivalent (Fig.2(c)(d)). As shown in Fig. 2(c), two systems (0=0.25a(X) and 0.25a(Y)) are essentially the same under 90° rotation. However, the peak heights of the commensurate oscillations in Xdisordered ALS systems are hardly affected at o=O.la and 0.25a (Fig.2(d)). It is found that the order of Ydirection is especially important for the commensurate oscillations. In the ALS systems which consisted of the rectangularly shaped anti-dots (11], the peak position of the commensurate oscillations depended on the direction of

297

Superlattices and Microstructures, Vol. 16. No .3. 1994 y

y

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024

024 Magnetic Field (kG)

Magnetic Field (kG) Fig . 2. (a)(c) Typical anti-dot arrays of ALS system in which the disorder is introduced only to the X- or V-dir ect ion. (b)( d) Magnetores istance of th e X-d isord ered and th e

V-disordered ALS systems. The anti-dots ar e arranged t riangularly (b) and squarely (d) . The peak height of the commensurate oscillations is affected by t he V-disord er more strongly than by th e X-disorder.

(a)

Square

~

ax

0.8 0 .6

ay= 1 u m d-0 .15 /l m

024 Magnetic Field (kG)

Fig. 3. (a ) Notation of periods a long the X- and the V-direction in square ALS system. (b)( c) Magnetoresistance of expanded ALS systems. (b) Distance of a y is fixed at 1 utti , and ax is expanded from 0.8 utt: to 3 J.lm. Th e magn etic field of the main peak is almost same . (c) 90° rotated ALS systems in

ax= 1 u m d-O .15 /lm 6

0 2 4 Magnetic Field (kG)

which ax is fixed at 1 utt: and a y is changed from 0.8 pm to 3 J.lm. The peak shifts drastically to low fields with increasi ng a y . Each peak almost agrees wit h the predicted magn etic field at which the cyclotron diameter is a y .

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Superlattices and Microstructures, Vol. 16, No.3, 1994

the current. The peak position of the commensurate oscillations and the direction of the current are closely connected.

3.2

Expanded ALS Systems

To confirm the importance of the order along the Ydirection, we investigated the expanded ALS systems to the X-direction or to the V-direction. The ordered 9000 anti-dots were arranged in each ALS system. In the squarely arranged ALS systems, the period a of the X- or the V-direction is defined in Fig. 3(a). Figure 3(b) presents the magnetoresistances of ALS systems with several ax at ay==1 Jim. The principal commensurate peak does not shift in spite of expansion of ax from 0.8 Jim to 3 Jim. However, in the 90' rotated ALS systems, in which ax is fixed at 1 Jim and a y is varied from 0.8 Jim to 3 Jim, the commensurate peak shifts drastically to low fields with increasing a y (Fig.3( c». Each magnetic field peak is in close agreement with the predicted magnetic field at which the cyclotron diameter is a y.

In the triangularly arranged ALS systems, however, the peak magnetic fields is not determined only by a y (FigA). The notations of the periods along the X- or the V-direction are shown in Fig. 4(a). In the fixed

ay(==1 Jim) ALS systems, the anti-dots form the equilateral triangle at ax= 0.866( =J3/2) Jim, and form the square at 0.5 Jim. The principal peak does not shift for a x= 0.866 ~ 3 Jim. In these three ALS systems (a y ~ Jax 2 + (ay/2)2), the nearest neighbor distance between anti-dots are the same (a y=1 Jim). In other ALS systems ofax= 0.683 Jim and 0.5 Jim (ay > Ja x2 + (ay/2)2), the peak shifts to higher magnetic field. In the squarely arranged ALS system (ax=0.5 Jim), the nearest neighbor distance is the side (l / J2 Jim) of square, not the diagonal (ay=l Jim). These results indicate that the nearest neighbor distance determines the commensurate condition. In the 90' rotated systems (Fig.4(c», the principal peak is also determined by the nearest neighbor distance. Without ALS system of a y=l Jim, the nearest neighbor distance is not a y but the diagonal (Ja x 2 + (ay/2)2). As a y becomes smaller, the nearest neighbor distance becomes monotonically smaller. As a result of the triangular ALS systems, the principal peak appears in the condition that the cyclotron diameter is commensurate to the nearest neighbor distance.

3.3

(a)

Period and Diameter The

magnetoresistance

(c)

also depends

T=1.5K Triangle

Triangle

ay=1/1 m

d-O.15.um

024 Magnetic Field (kG)

a x=1/1m

d-O.15tL m

024 Magnetic Field (kG)

Fig. 4. (a) Notation of periods along the X- and the V-direction in triangular ALS system. (b)(c) Magnetoresistance of expanded ALS systems. (b) At a y=1 Jim, the anti-dots form the equilateral triangle (a x= 0.866(=J3/2) Jim), and the square (a x=0.5 Jim). The main peak does not shift between a x= 0.866 Jim and 3 Jim. The nearest neighbor distance is the same (1 Jim). However, it shifts to higher magnetic field at a x= 0.683 Jim and 0.5 Jim. The nearest neighbor distance is the side of square. (c) In the 90' rotated systems, the peak is also determined by the nearest neighbor distance.

on

the

299

Superlattices and Microstructures, Vol. 16, No.3, 1994 ALS period (1 ~ 4 /lm) and the anti-dot diameter (0.2 ~ 1.2 /lm) in the ordered triangular ALS systems (Fig.5). Each area is 200 /lm x 150 /lm. The typical anti-dot arrays are shown in the upper illustration of Fig. 5(a) and 5(b). If the peaks of the magnetoresistance is determined by the commensurate condition between a and the cyclotron radius (Rc), the magnetoresistance would be the same in the normalized magnetic field (a/2Rc). With increasing period, the principal peak magnetic fields (a/2Rc) slightly shift to higher field (Fig.5(a)). Because of the limited mean free path of electrons (12 /lm), electrons with the large cyclotron orbits would be scattered in the orbit, and the contribution of the lower field in the peak would become weak. The principal peak shifts to the higher field as the period becomes larger. At a=1 /lm, the peak encircling 7 anti-dots clearly appears, but the peak encircling 3 antidots does not. However, at a=2 um, the situation is reverse. In addition, the peak heights also drastically depend on the anti-dot diameter (Fig.5(b)). At d~0.8 /lm, the magnetoresistance resembles that of a=1 /lm at d~0.2 /lm (Fig.5(a): uppermost magnetoresistance). The peak encircling 3 anti-dots appears clearly when the ratio of the anti-dot diameter to the ALS period (d/a) is small, while, the peak encircling 7 anti-dots appears when the ratio is large. The magnetoresistance peaks due

(a)

104

to the commensurate oscillations can not be understood without the consideration of its period and diameter.

3.4

The negative magnetoresistance in the very low magnetic field (less than 100 Gauss) was observed both in the ordered and in the disordered ALS system. The negative magnetoresistance was not observed in magnetic field parallel to the plane of the 2DEG. Because of temperature dependence of low field magnetoresistance (shown in Fig.6(a) inset, three symbols) and the disappearance above 10 K, we fitted the negative magnetoresistance following the two dimensional weak localization theory of Hikami-Larkin-Nagaoka [12) (Fig.6(a) inset, solid line). The estimated inelastic scattering length from the fitting is about 1 utt: at 0.4 K, and decreases with increasing temperature. Since the apparent difference between ALS systems with different (T is not observed, the negative magnetoresistance may be caused by the defects around the anti-dots. With decreasing temperature below 20 ~ 40 K, the zero field conductance decreases, while, the conductance of the 2DEG (without anti-dots) region increases. In the two dimensional weak localization theory, the relation between the temperature and the conductance change is

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1

2

Magnetic Field (kG)

Fig. 5. Magnetoresistance for various anti-dots arrays. Period dependence (a), and diameter dependence (b). R c is cyclotron radius.

3

Superlattices and Microstructures, Vol. 16, No.3, 1994

300

a=0.85J1 m

oo s:

OJ

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0.8

OJ

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8

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100

Fig. 6. (a) Inelastic scattering length in the ordered and the disordered ALS systems. Inset: Magnetoconductance of (J = 0.25a in very low magnetic field for differing temperature with the theoretical curves (solid lines). (b) Temperature dependence of the conductance change in the absence of magnetic field.

expressed as /:}.(J = OT( e 2 /21r 2h )lnT [13]. In our experiments, the value of the coefficient OT is 3 ~6. This is much larger than OT ~1 which is expected by the conventional week Anderson localization [13, 14J. The origin of the temperature dependence in the high temperature region have not yet been revealed.

4

Conclusion

We have investigated the transport properties in the various ALS systems. In the disordered ALS systems, the peak heights of the commensurate oscillations decrease with increasing the shift of the anti-dot location. Especially, the peak height is affected by the disorder along the Y-direction (perpendicular to the current direction) more strongly than by the disorder along the X-direction. In the expanded square ALS systems to the X-direction or the Y-direction, the commensurate oscillations should be determined by the order along the Y-direction. However, in the expanded triangular ALS systems, the peak magnetic field is determined by the nearest neighbor distance. In addition, each peak height depends on the ratio of the anti-dot diameter to the ALS period. The negative magnetoresistance in very low magnetic field (less than 100 Gauss) is observed both in the ordered and the disordered ALS systems due to the two dimensional weak localization effect. Moreover, the large temperature dependence of the conductance, which can

not be explained by the conventional localization theory, is observed. Acknowledgement We would like to thank H.Okada of Sumitomo Electric Industry Co. ,Ltd. for providing high mobility heterostructure, and J.Takahara, T.Nakanishi and T.Ohtsuki for valuable discussions. This work is partially supported by Grant-in-Aid for Scientific Research (8) and Scientific Research on Priority Area from the Ministry of Education, Science and Culture (Japan). The author (K.T.) acknowledge for supporting by JSPS Research Fellowships for Young Scientists. One of the authors (S.T.) acknowledge the support by Casio Science Promotion and Shimazu Science Foundation.

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[9] G. Berthold, J. Smoliner, V. Rosskopf, E. Gornik, G. Bohrn, and G. Weimann, Phys. Rev. B47, 10383 (1993). [10] C. T. Liang, C. G. Smith, J. T. Nicholls, R. J. F. Hughes, M. Pepper, J. E. F. Frost, D. A. Ritchie, M. P. Grimshaw, and G. A. C. Jones, Phys. Rev. B49, 8518 (1994). [11] R. Schuster, K. Ensslin, J. P. Kotthaus, M. Holland, and C. Stanley, Phys. Rev. B47, 6843 (1993). [12] S. Hikami, A. 1. Larkin, and Y. Nagaoka, Prog. Theor. Phys. 63, 707 (1980). [13] D. J. Bishop, D. C. Tsui, and R. C. Dynes, Phys. Rev. Lett. 44, 1153 (1980). [14] T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54,437 (1982).